Geometry II

Volume 29 of the series Encyclopaedia of Mathematical Sciences pp 139-248

Discrete Groups of Motions of Spaces of Constant Curvature

  • E. B. Vinberg
  • , O. V. Shvartsman

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Discrete groups of motions of spaces of constant curvature, as well as other groups that can be regarded as such (although they may be defined differently), arise naturally in different areas of mathematics and its applications. Examples are the symmetry groups of regular polyhedra, symmetry groups of ornaments and crystallographic structures, discrete groups of holomorphic transformations arising in the uniformization theory of Riemannian surfaces, fundamental groups of space forms, groups of integer Lorentz transformations etc. (see Chapter 1). Their study fills a brilliant page in the development of geometry.